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INTRODUCTION 3

CHAPTER Ⅰ Clifford Algebras，Spin Groups and Their Representations 7

1．Clifford algebras 7

2．The groups Pin and Spin 12

3．The algebras Clnand Clr,s 20

4．The classification 25

5．Representations 30

6．Lie algebra structures 40

7．Some direct applications to geometry 44

8．Some further applications to the theory of Lie groups 49

9．K-theory and the Atiyah-Bott-Shapiro construction 58

10．KR-theory and the (1,1)-Periodicity Theorem 70

CHAPTER Ⅱ Spin Geometry and the Dirac Operators 77

1．Spin structures on vector bundles 78

2．Spin manifolds and spin cobordism 85

3．Clifford and spinor bundles 93

4．Connections on spinor bundles 101

5．The Dirac operators 112

6．The fundamental elliptic operators 135

7．Clk-linear Dirac operators 139

8．Vanishing theorems and some applications 153

CHAPTER Ⅲ Index Theorems 166

1．Differential operators 167

2．Sobolev spaces and Sobolev theorems 170

3．Pseudodifferential operators 177

4．Elliptic operators and parametrices 188

5．Fundamental results for elliptic operators 192

6．The heat kernel and the index 198

7．The topological invariance of the index 201

8．The index of a family of elliptic operators 205

9．The G-index 211

10．The Clifford index 214

11．Multiplicative sequences and the Chern character 225

12．Thom isomorphisms and the Chern character defect 238

13．The Atiyah-Singer Index Theorem 243

14．Fixed-point formulas for elliptic operators 259

15．The Index Theorem for Families 268

16．Families of real operators and the Clk-index Theorem 270

17．Remarks on heat and supersymmetry 277

CHAPTER Ⅳ Applications in Geometry and Topology 278

1．Integrality theorems 280

2．Immersions of manifolds and the vector field problem 281

3．Group actions on manifolds 291

4．Compact manifolds of positive scalar curvature 297

5．Positive scalar curvature and the fundamental group 302

6．Complete manifolds of positive scalar curvature 313

7．The topology of the space of positive scalar curvature metrics 326

8．Clifford multiplication and K？hler manifolds 330

9．Pure spinors,complex structures，and twistors 335

10．Reduced holonomy and calibrations 345

11．Spinor cohomology and complex manifolds with vanishing first Chern class 357

12．The Positive Mass Conjecture in general relativity 368

APPENDIX A Principal G-bundles 370

APPENDIX B Classifying Spaces and Characteristic Classes 376

APPENDIX C Orientation Classes and Thom Isomorphisms in K-theory 384

APPENDIX D Spinc-manifolds 390

BIBLIOGRAPHY 402

INDEX 417

NOTATION INDEX 425